The present invention relates in general to mass flow measuring techniques and, in particular to a new and useful apparatus and method of measuring mass flow rate of the fluid utilizing two spaced apart tubes each meant for carrying about one half of the flow, which tubes are forced to oscillate between fixed points in order to impart a reciprocating angular rotation to the tubes.
Devices are known which utilize the effect of angular motion on a moving field to directly measure mass flow. See for example, U.S. Pat. No. 2,865,201 issued Dec. 23, 1958 to Roth and U.S. Pat. No. 3,355,944 issued Dec. 5, 1967 and 3,485,098 issued Dec. 23, 1969 to Sipin.
U.S. Pat. No. 4,109,524 issued Aug. 29, 1978 to Smith, discloses an apparatus and method for measuring mass flow rate through a conduit by reciprocating a section of the conduit to produce longitudinal angular rotation of that section. Linkages are connected to the section both for reciprocating it and for measuring a force exerted on the section which force is due to an apparent force produced by mass flow through the conduit section. A direct measurement can thus be taken of the mass flow rate in this manner.
To understand how mass flow rate can be measured using the effects of this force, reference is now made to FIG. 1 which shows an arrangement of vectors on an X, Y, coordinate system.
In order for a mass (m), which is moving with velocity (v), to maintain an angular velocity (w) about some axis perpendicular to v, it must be subjected to a coriolis force (F.sub.c) which is perpendicular to both v and w.
The coriolis acceleration can be derived in a straightforward and simple fashion from the time derivitives of the transformation between local rectangular (x,y) and polar (r, .phi.) coordinate systems as shown in FIG. 2. EQU x=r cos .phi. EQU y=r sin .phi. EQU x'=-r.phi.' sin .phi.+r' cos .phi. EQU y'=r.phi.' cos .phi.+r' sin .phi. EQU x"=[r"-r.phi.'.sup.2 ] cos .phi.-[r.phi."+2r'.phi.'] sin .phi. EQU y"=[r"-r.phi.'.sup.2 ] sin .phi.+[r.phi."+2r'.phi.'] cos .phi.
The "2r'.phi.'" component is the coriolis acceleration. Note that it is the product of two velocities, and that it is not a function of the distance (r) from the origin. Thus it must also exist at the origin (r=0), and this helps in understanding how or why the mass flowmeter works. For example, one can visualize a local origin at each point along a tube. As the tube flexes, each point rotates to some extent. If fluid is also flowing through the tube (r'), then this rotation of the tube (.phi.') causes a tangetial coriolis acceleration of (2r'.phi.').
For fluid flowing from left to right in a tube aligned with the x-axis (.phi.=0), the coriolis acceleration is in the y-direction and may be written as follows. EQU y.sub.c "=2V.sub.f .phi.'
where:
y(x,t)=displacement function of the tube PA0 V.sub.f =velocity of the fluid=r' PA0 .phi.=dy/dx (partials) PA0 .phi.'=d.sup.2 y/dt dx (partials)
The inertial force exerted on the tube by the coriolis acceleration of the fluid inside the tube for element length dx is as follows. ##EQU1##
This is an unsymmetric relation which is one reason for its utility in this device, i.e., a rotational velocity produces a translational force, but the translational velocity does not produce a rotational force. Thus, if symmetric modes of the mass flowmeter tubes are driven by external means such as electromagnetic shakers, the coriolis forces tend to excite the anti-symmetric modes. Conversely, if anti-symmetric modes are externally driven, the coriolis forces tend to excite the symmetric modes.
If the driving force is sinusoidal, then the tube's displacement, velocity and acceleration will likewise be sinusoidal and vary by 90.degree. and 180.degree. respectively. This allows the phase difference .phi. to be equal regardless of whether it is measured relative to the displacement, velocity or acceleration functions of the drive force versus resultant drive force plus the force --F.sub.c.